Ybus Formation Algorithm

• Step 1: Initialize Ybus as zero matrix of size n×n

• Step 2: For each transmission line between buses i and j:

  • Add line admittance \(y_{ij}\) to \(Y_{ii}\) and \(Y_{jj}\)

  • Subtract line admittance \(y_{ij}\) from \(Y_{ij}\) and \(Y_{ji}\)

• Step 3: For each shunt element at bus i:

  • Add shunt admittance \(y_i\) to \(Y_{ii}\)

• Step 4: For transformers, modify admittances based on tap settings

Ybus Modification Techniques

• Adding a New Bus:

  • Increase matrix size to \((n+1) \times (n+1)\)

  • Add new row and column

  • Update diagonal elements for connected buses

• Changing a Line Impedance:

  • Calculate difference: \(\Delta y = y_{new} - y_{old}\)

  • Modify: \(Y_{ii} = Y_{ii} + \Delta y\), \(Y_{jj} = Y_{jj} + \Delta y\)

  • Modify: \(Y_{ij} = Y_{ij} - \Delta y\), \(Y_{ji} = Y_{ji} - \Delta y\)

• Removing a Line:

  • Subtract old line admittance from diagonal elements

  • Add old line admittance to off-diagonal elements

Load Flow Problem Formulation

• Power Flow Equations:

  \[P_i = |V_i| \sum_{j=1}^n |V_j| |Y_{ij}| \cos(\theta_{ij} + \delta_j - \delta_i)\]
  \[Q_i = -|V_i| \sum_{j=1}^n |V_j| |Y_{ij}| \sin(\theta_{ij} + \delta_j - \delta_i)\]

• Bus Classifications:

  • Slack Bus: \(|V|\) and \(\delta\) specified

  • PV Bus: \(P\) and \(|V|\) specified

  • PQ Bus: \(P\) and \(Q\) specified

• Unknowns: \((n-1)\) voltage angles + \((n-m-1)\) voltage magnitudes

• Equations: \((n-1)\) active power + \((n-m-1)\) reactive power

Gauss-Seidel Method

• Iterative Formula:

  \[V_i^{k+1} = \dfrac{1}{Y_{ii}} \left[ \dfrac{P_i - jQ_i}{(V_i^k)^*} - \sum_{j=1}^{i-1} Y_{ij} V_j^{k+1} - \sum_{j=i+1}^n Y_{ij} V_j^k \right]\]

• For PV Buses:

  • Calculate \(Q_i\) from voltage update

  • If \(Q_i\) within limits, use calculated \(V_i^{k+1}\)

  • If \(Q_i\) exceeds limits, treat as PQ bus

• Acceleration Factor:

  \[V_i^{k+1} = V_i^k + \alpha(V_i^{calc} - V_i^k)\]
  where \(\alpha = 1.3\) to \(1.6\) for optimal convergence

Newton-Raphson Method

• Power Mismatch Equations:

  \[\Delta P_i = P_i^{spec} - P_i^{calc}\]
  \[\Delta Q_i = Q_i^{spec} - Q_i^{calc}\]

• Linearized System:

  \[\begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta \delta \\ \Delta |V| \end{bmatrix}\]

• Jacobian Elements:

  • \(J_1 = \dfrac{\partial P}{\partial \delta}\), \(J_2 = \dfrac{\partial P}{\partial |V|}\)

  • \(J_3 = \dfrac{\partial Q}{\partial \delta}\), \(J_4 = \dfrac{\partial Q}{\partial |V|}\)

Jacobian Matrix Elements

• Diagonal Elements:

  \[J_{1ii} = \dfrac{\partial P_i}{\partial \delta_i} = -Q_i - |V_i|^2 B_{ii}\]
  \[J_{4ii} = \dfrac{\partial Q_i}{\partial |V_i|} = \dfrac{Q_i}{|V_i|} + |V_i| B_{ii}\]

• Off-Diagonal Elements:

  \[J_{1ij} = \dfrac{\partial P_i}{\partial \delta_j} = |V_i||V_j|[G_{ij}\sin(\delta_i-\delta_j) - B_{ij}\cos(\delta_i-\delta_j)]\]
  \[J_{2ij} = \dfrac{\partial P_i}{\partial |V_j|} = |V_i|[G_{ij}\cos(\delta_i-\delta_j) + B_{ij}\sin(\delta_i-\delta_j)]\]

Comparison of Load Flow Methods

Fast Decoupled Load Flow

• Decoupling Assumptions:

  • \(\dfrac{\partial P}{\partial |V|} \approx 0\) (weak coupling)

  • \(\dfrac{\partial Q}{\partial \delta} \approx 0\) (weak coupling)

  • \(Q_i \ll |V_i|^2 B_{ii}\) (high X/R ratio)

• Simplified Equations:

  \[\Delta P = B' \Delta \delta\]
  \[\Delta Q = B'' \Delta |V|\]

• Advantages:

  • Faster than Newton-Raphson

  • Constant coefficient matrices

  • Single matrix factorization

Voltage Control - Basic Concepts

• Voltage Regulation:

  \[\text{Voltage Regulation} = \dfrac{V_{no-load} - V_{full-load}}{V_{full-load}} \times 100\%\]

• Reactive Power - Voltage Relationship:

  \[V_r = V_s - \dfrac{XQ}{V_s}\]
  where \(V_r\) = receiving end voltage, \(V_s\) = sending end voltage

• Control Objectives:

  • Maintain voltage within \(\pm 5\%\) of nominal

  • Minimize transmission losses

  • Improve system stability

Voltage Control Methods

Frequency Control - Basic Concepts

• Frequency Deviation:

  \[\Delta f = f - f_0\]
  where \(f_0 = 50\) Hz (nominal frequency)

• Power-Frequency Relationship:

  \[\Delta P = D \Delta f\]
  where \(D\) is load damping coefficient

• Control Objectives:

  • Maintain frequency within \(\pm 0.5\) Hz of nominal

  • Balance generation and load continuously

  • Minimize area control error (ACE)

Frequency Control Hierarchy

• Primary Control (Governor Response):

  • Response time: 5-10 seconds

  • Droop characteristic: 4-5%

  • Local control action

  • Arrests frequency deviation

• Secondary Control (AGC):

  • Response time: 30 seconds to 1 minute

  • Restores frequency to nominal

  • Maintains scheduled tie-line flows

  • Centralized control

• Tertiary Control (Economic Dispatch):

  • Response time: 15-30 minutes

  • Optimizes generation costs

  • Manages reserves

  • Manual/automatic operation

Governor Characteristics

• Droop Characteristic:

  \[R = \dfrac{\Delta f / f_0}{\Delta P / P_{rated}} \times 100\%\]

• Speed Regulation:

  \[R = \dfrac{f_{nl} - f_{fl}}{f_{fl}} \times \dfrac{P_{rated}}{P_{fl}} \times 100\%\]

• Free Governor Mode:

  • Generator follows load changes

  • Frequency varies with load

• Isochronous Mode:

  • Maintains constant frequency

  • Used for single machine systems

Load Frequency Control (LFC)

• Single Area System:

  • Transfer function: \(G(s) = \dfrac{1}{Ds + 1/R}\)

  • Time constant: \(T = \dfrac{H}{D}\)

  • Frequency response: \(\Delta f = \dfrac{-\Delta P_L}{D + 1/R}\)

• Multi-Area System:

  • Area Control Error: \(ACE_i = \Delta P_{tie,i} + B_i \Delta f_i\)

  • Frequency bias: \(B_i = \dfrac{1}{R_i} + D_i\)

  • Tie-line power: \(\Delta P_{tie} = \dfrac{2\pi T_{12}}{s}(\Delta f_1 - \Delta f_2)\)

Automatic Generation Control (AGC)

• Objectives:

  • Regulate frequency to scheduled value

  • Maintain net interchange at scheduled value

  • Distribute load among units economically

• Control Strategies:

  • Proportional control: \(\Delta P_c = -K_p \times ACE\)

  • Integral control: \(\Delta P_c = -K_i \int ACE \, dt\)

  • PI control: \(\Delta P_c = -K_p \times ACE - K_i \int ACE \, dt\)

• Performance Indices:

  • Settling time

  • Maximum frequency deviation

  • Steady-state error

Key Formulas for GATE

1. Ybus diagonal element: \(Y_{ii} = \sum_{k=1}^n y_{ik}\)

2. Gauss-Seidel voltage update:

   \[V_i^{k+1} = \dfrac{1}{Y_{ii}} \left[ \dfrac{P_i - jQ_i}{(V_i^k)^*} - \sum_{j\neq i} Y_{ij} V_j \right]\]

3. Power flow equations:

   \[P_i = |V_i| \sum_{j=1}^n |V_j| |Y_{ij}| \cos(\theta_{ij} + \delta_j - \delta_i)\]

4. Governor droop: \(R = \dfrac{\Delta f / f_0}{\Delta P / P_{rated}} \times 100\%\)

5. Area Control Error: \(ACE = \Delta P_{tie} + B \Delta f\)

6. Frequency response: \(\Delta f_{ss} = \dfrac{-\Delta P_L}{D + 1/R}\)

Important GATE Points

• Ybus Properties:

  • Always singular (determinant = 0)

  • Symmetric for passive networks

  • Sparse matrix structure

• Load Flow Convergence:

  • Newton-Raphson: 3-5 iterations

  • Gauss-Seidel: 50-100 iterations

  • Flat start: \(V = 1.0 \angle 0°\) for all buses

• Control Hierarchy:

  • Primary: Governor (seconds)

  • Secondary: AGC (minutes)

  • Tertiary: Economic dispatch (hours)